Transformation methods of wavefront maps from one vertex distance to another

ABSTRACT

The present invention provides methods, systems and software for scaling optical aberration measurements of optical systems. In one embodiment, the present invention provides a method of reconstructing optical tissues of an eye. The method comprises transmitting an image through the optical tissues of the eye. Aberration data from the transmitted image is measured across the optical tissues of the eye at a first plane. A conversion algorithm is applied to the data, converting it to corrective optical power data that can be used as a basis for constructing a treatment for the eye at a second plane.

CROSS-REFERENCE TO RELATED APPLICATIONS

This is a nonprovisional patent application which claims the benefitunder 35 USC 119(e) of U.S. Provisional Patent Application No.60/550,514 filed Mar. 3, 2004, the full disclosure of which isincorporated herein by reference.

BACKGROUND OF THE INVENTION

The present invention generally relates to scaling optical aberrationmeasurements of optical systems. More particularly, the inventionrelates to improved methods and systems for processing optical powermeasurements taken at a first plane and converting those powermeasurements to corrective optical power measurements that can be usedat a second plane. The present invention may be useful in any of avariety of ocular treatment modalities, including ablative laser eyesurgery, contact lenses, spectacles, intraocular lenses, and the like.

Known laser eye surgery procedures generally employ an ultraviolet orinfrared laser to remove a microscopic layer of stromal tissue from thecornea of the eye. The laser typically removes a selected shape of thecorneal tissue, often to correct refractive errors of the eye.Ultraviolet laser ablation results in photodecomposition of the cornealtissue, but generally does not cause significant thermal damage toadjacent and underlying tissues of the eye. The irradiated molecules arebroken into smaller volatile fragments photochemically, directlybreaking the intermolecular bonds.

Laser ablation procedures can remove the targeted stroma of the corneato change the cornea's contour for varying purposes, such as forcorrecting myopia, hyperopia, astigmatism, and the like. Control overthe distribution of ablation energy across the cornea may be provided bya variety of systems and methods, including the use of ablatable masks,fixed and moveable apertures, controlled scanning systems, eye movementtracking mechanisms, and the like. In known systems, the laser beamoften comprises a series of discrete pulses of laser light energy, withthe total shape and amount of tissue removed being determined by theshape, size, location, and/or number of laser energy pulses impinging onthe cornea. A variety of algorithms may be used to calculate the patternof laser pulses used to reshape the cornea so as to correct a refractiveerror of the eye. Known systems make use of a variety of forms of lasersand/or laser energy to effect the correction, including infrared lasers,ultraviolet lasers, femtosecond lasers, wavelength multipliedsolid-state lasers, and the like. Alternative vision correctiontechniques make use of radial incisions in the cornea, intraocularlenses, removable corneal support structures, and the like.

Known corneal correction treatment methods have generally beensuccessful in correcting standard vision errors, such as myopia,hyperopia, astigmatism, and the like. However, as with all successes,still further improvements would be desirable. Toward that end,wavefront measurement systems are now available to accurately measurethe refractive characteristics of a particular patient's eye. Oneexemplary wavefront technology system is the VISX WaveScan® System,which uses a Hartmann-Shack wavefront lenslet array that can quantifyaberrations throughout the entire optical system of the patient's eye,including first- and second-order sphero-cylindrical errors, coma, andthird and fourth-order aberrations related to coma, astigmatism, andspherical aberrations.

Wavefront measurement of the eye may be used to create a high orderaberration map or wavefront elevation map that permits assessment ofaberrations throughout the optical pathway of the eye, e.g., bothinternal aberrations and aberrations on the corneal surface. Theaberration map may then be used to compute a custom ablation pattern forallowing a surgical laser system to correct the complex aberrations inand on the patient's eye. Known methods for calculation of a customizedablation pattern using wavefront sensor data generally involvesmathematically modeling an optical surface of the eye using expansionseries techniques. More specifically, Zernike polynomials have beenemployed to model the optical surface, as proposed by Liang et al., inObjective Measurement of Wave Aberrations of the Human Eye with the Useof a Harman-Shack Wave-front Sensor, Journal Optical Society of America,July, 1994, vol. 11, No. 7, pp. 1949-1957, the entire contents of whichis hereby incorporated by reference. Coefficients of the Zernikepolynomials are derived through known fitting techniques, and therefractive correction procedure is then determined using the shape ofthe optical surface of the eye, as indicated by the mathematical seriesexpansion model.

Optical measurements such as wavefront measurements are often taken at ameasurement plane, whereas optical treatments may be needed at atreatment plane that is different from the measurement plane. Thus,power adjustments are often used when devising optical treatments forpatients. For example, power adjustments can be used by optometristswhen prescribing spectacles for patients. Typically, refractivemeasurements are made by an optometer at a measurement plane somedistance anterior to the eye, and this distance may not coincide withthe spectacle plane. Thus, the measured power corresponding to themeasurement plane may need to be converted to a corrective powercorresponding to the spectacle or treatment plane. Similarly, whenwavefront measurements are obtained with wavefront devices, in manycases the measured map is conjugated at the pupil plane, which is notthe same as the corneal plane or spectacle plane. To enhance theeffectiveness of a refractive surgical procedure, vertex correction maybe needed to adjust the power of the measured maps. Yet there remains alack of efficient methods and systems for such power conversions.

Therefore, in light of above, it would be desirable to provide improvedmethods and systems for processing optical data taken at a measurementplane and converting that optical data to corrective optical data thatcan be used at a treatment plane.

BRIEF SUMMARY OF THE INVENTION

The present invention provides methods and systems for processingoptical power measurements taken at a first plane and converting thosepower measurements to corrective optical power measurements that can beused at a second plane.

In one aspect, the present invention provides a method of determining arefractive treatment shape for ameliorating a vision condition in apatient. The method comprises measuring a wavefront aberration of an eyeof the patient in order to provide a measurement surface aberration,deriving a treatment surface aberration of the eye based on themeasurement surface aberration, and determining the refractive treatmentshape based on the treatment surface aberration of the eye. Thewavefront aberration can correspond to a measurement surface that isdisposed at or near a pupil plane of the eye, and the treatment surfaceaberration can correspond to a treatment surface that is disposed at ornear an anterior surface of a cornea of the eye. The treatment surfaceaberration may be derived using a difference between the measurementsurface and the treatment surface.

In another aspect, the present invention provides a method ofameliorating a vision condition in a patient. The method comprisesmeasuring a wavefront aberration of an eye of the patient in order toprovide a measurement surface aberration, deriving a treatment surfaceaberration of the eye from the measurement surface aberration,determining a refractive treatment shape based on the treatment surfaceaberration of the eye, and applying the refractive treatment shape tothe eye of the patient to ameliorate the vision condition. The wavefrontaberration can correspond to a measurement surface that is disposed ator near a pupil plane of the eye. The treatment surface aberration cancorrespond to a treatment surface that is disposed at or near ananterior corneal surface of the eye, or a treatment surface thatcorresponds to a spectacle plane of the eye. Relatedly, the treatmentsurface may be disposed posterior to a pupil plane of the eye. Thetreatment surface aberration may be based on a difference between themeasurement surface and the treatment surface.

In a related aspect, the refractive treatment shape can be applied tothe eye of the patient in a variety of treatment modalities. Forexample, the treatment shape can be applied by ablating a cornealsurface of the patient to provide a corneal surface shape thatcorresponds to the refractive treatment shape. The treatment shape mayalso be applied by providing the patient with a contact lens that has ashape that corresponds to the refractive treatment shape. Further, thetreatment shape may be applied by providing the patient with a spectaclelens that has a shape that corresponds to the refractive treatmentshape. What is more, the treatment shape can be applied by providing thepatient with an intra-ocular lens that has a shape that corresponds tothe refractive treatment shape.

In another aspect, the present invention provides a system forgenerating a refractive treatment shape for ameliorating a visioncondition in an eye of a patient. The system comprises an input modulethat accepts a measurement surface aberration, a transformation modulethat derives a treatment surface aberration based on the measurementsurface aberration, and an output module that generates the refractivetreatment shape based on the treatment surface aberration. Themeasurement surface aberration may be based on a wavefront aberration ofthe eye. The wavefront aberration can correspond to a measurementsurface that is disposed at or near a pupil plane of the eye. Thetreatment surface aberration can correspond to a treatment surface thatis disposed at or near an anterior corneal surface of the eye, or atreatment surface that corresponds to a spectacle plane of the eye.Relatedly, the treatment surface may be disposed posterior to a pupilplane of the eye. The treatment surface aberration may be based on adifference between the measurement surface and the treatment surface.

In another aspect, the present invention provides a system forameliorating a vision condition in an eye of a patient. The systemcomprises an input module that accepts a measurement surface aberration,a transformation module that derives a treatment surface aberrationbased on the measurement surface aberration, an output module thatgenerates a refractive treatment shape based on the treatment surfaceaberration, and a laser system that directs laser energy onto the eyeaccording to the refractive treatment shape so as to reprofile a surfaceof the eye from an initial shape to a subsequent shape, the subsequentshape having correctively improved optical properties for amelioratingthe vision condition. The measurement surface aberration may be based ona wavefront aberration of the eye. The wavefront aberration cancorrespond to a measurement surface that is disposed at or near a pupilplane of the eye, and the treatment surface aberration can correspond toa treatment surface that is disposed at or near an anterior surface of acornea of the eye. The treatment surface aberration can be derived basedon a difference between the measurement surface and the treatmentsurface.

In some aspects, the treatment surface aberration may be a treatmentsurface wavefront map. In other aspects, the measurement surfaceaberration may be a measurement surface wavefront map. The treatmentsurface wavefront map may be derived at least in part by local slopescaling of the measurement surface wavefront map. In still otheraspects, the treatment surface wavefront map may be derived at least inpart by applying a scaling factor of 1/(1+Pd) to a slope of themeasurement surface wavefront map, where P represents a local curvatureof the measurement surface wavefront map and d represents a differencebetween the measurement surface and the treatment surface. In a relatedaspect, a difference between the measurement surface and a retinalsurface of the eye corresponds to a first vertex measure, and adifference between the treatment surface and the retinal surface of theeye corresponds to a second vertex measure. P may be based on a secondderivative of the measurement surface wavefront map. P may also be basedon a pupil radius of the eye.

In some aspects, the treatment surface wavefront map can be derivedaccording to an iterative Fourier reconstruction algorithm. What ismore, the measurement surface aberration may reflect low order and/orhigh order aberrations of the eye of the patient.

In another aspect, the present invention provides a system forgenerating a prescription for ameliorating a vision condition in an eyeof a patient. The system comprises an input that accepts irregularaberration data corresponding to an aberration measurement surfaceadjacent a pupil plane of the eye, a transformation module that derivesa treatment surface aberration corresponding to a treatment surface thatis disposed adjacent an anterior surface of a cornea of the eye, and anoutput module that generates the prescription based on the treatmentsurface aberration. The treatment surface aberration can be derived fromthe irregular aberration data using a difference between the measurementsurface and the treatment surface.

These and other aspects will be apparent in the remainder of thefigures, description, and claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a laser ablation system according to an embodiment ofthe present invention.

FIG. 2 illustrates a simplified computer system according to anembodiment of the present invention.

FIG. 3 illustrates a wavefront measurement system according to anembodiment of the present invention.

FIG. 3A illustrates another wavefront measurement system according to anembodiment of the present invention.

FIG. 4 schematically represents a simplified set of modules that carryout one method of the present invention.

FIG. 5 is a flow chart that schematically illustrates a method ofdetermining a refractive treatment shape according to one embodiment ofthe present invention.

FIG. 6 illustrates a model optical system.

FIG. 7 illustrates a comparison between vertex corrected powercalculations based on algorithms provided by the present invention withcalculations based on a classical formula.

FIG. 8 illustrates a wavefront before and after a vertex correction.

DETAILED DESCRIPTION OF THE INVENTION

The present invention provides methods, software, and systems forprocessing optical power measurements taken at a first plane andconverting those power measurements to corrective optical powermeasurements that can be used at a second plane.

The present invention is generally useful for enhancing the accuracy andefficacy of laser eye surgical procedures, such as photorefractivekeratectomy (PRK), phototherapeutic keratectomy (PTK), laser in situkeratomileusis (LASIK), and the like. The present invention can provideenhanced optical accuracy of refractive procedures by improving themethodology for processing measured optical errors of the eye and hencecalculate a more accurate refractive ablation program. In one particularembodiment, the present invention is related to therapeuticwavefront-based ablations of pathological eyes.

The present invention can be readily adapted for use with existing lasersystems, wavefront measurement systems, and other optical measurementdevices. While the systems, software, and methods of the presentinvention are described primarily in the context of a laser eye surgerysystem, it should be understood the present invention may be adapted foruse in alternative eye treatment procedures and systems such asspectacle lenses, intraocular lenses, contact lenses, corneal ringimplants, collagenous corneal tissue thermal remodeling, and the like.

Turning now to the drawings, FIG. 1 illustrates a laser eye surgerysystem 10 of the present invention, including a laser 12 that produces alaser beam 14. Laser 12 is optically coupled to laser delivery optics16, which directs laser beam 14 to an eye E of patient P. A deliveryoptics support structure (not shown here for clarity) extends from aframe 18 supporting laser 12. A microscope 20 is mounted on the deliveryoptics support structure, the microscope often being used to image acornea of eye E.

Laser 12 generally comprises an excimer laser, ideally comprising anargon-fluorine laser producing pulses of laser light having a wavelengthof approximately 193 nm. Laser 12 will preferably be designed to providea feedback stabilized fluence at the patient's eye, delivered viadelivery optics 16. The present invention may also be useful withalternative sources of ultraviolet or infrared radiation, particularlythose adapted to controllably ablate the corneal tissue without causingsignificant damage to adjacent and/or underlying tissues of the eye.Such sources include, but are not limited to, solid state lasers andother devices which can generate energy in the ultraviolet wavelengthbetween about 185 and 215 nm and/or those which utilizefrequency-multiplying techniques. Hence, although an excimer laser isthe illustrative source of an ablating beam, other lasers may be used inthe present invention.

Laser system 10 will generally include a computer or programmableprocessor 22. Processor 22 may comprise (or interface with) aconventional PC system including the standard user interface devicessuch as a keyboard, a display monitor, and the like. Processor 22 willtypically include an input device such as a magnetic or optical diskdrive, an internet connection, or the like. Such input devices willoften be used to download a computer executable code from a tangiblestorage media 29 embodying any of the methods of the present invention.Tangible storage media 29 may take the form of a floppy disk, an opticaldisk, a data tape, a volatile or non-volatile memory, RAM, or the like,and the processor 22 will include the memory boards and other standardcomponents of modem computer systems for storing and executing thiscode. Tangible storage media 29 may optionally embody wavefront sensordata, wavefront gradients, a wavefront elevation map, a treatment map, acorneal elevation map, and/or an ablation table. While tangible storagemedia 29 will often be used directly in cooperation with a input deviceof processor 22, the storage media may also be remotely operativelycoupled with processor by means of network connections such as theinternet, and by wireless methods such as infrared, Bluetooth, or thelike.

Laser 12 and delivery optics 16 will generally direct laser beam 14 tothe eye of patient P under the direction of a computer 22. Computer 22will often selectively adjust laser beam 14 to expose portions of thecornea to the pulses of laser energy so as to effect a predeterminedsculpting of the cornea and alter the refractive characteristics of theeye. In many embodiments, both laser beam 14 and the laser deliveryoptical system 16 will be under computer control of processor 22 toeffect the desired laser sculpting process, with the processor effecting(and optionally modifying) the pattern of laser pulses. The pattern ofpulses may by summarized in machine readable data of tangible storagemedia 29 in the form of a treatment table, and the treatment table maybe adjusted according to feedback input into processor 22 from anautomated image analysis system in response to feedback data providedfrom an ablation monitoring system feedback system. Optionally, thefeedback may be manually entered into the processor by a systemoperator. Such feedback might be provided by integrating the wavefrontmeasurement system described below with the laser treatment system 10,and processor 22 may continue and/or terminate a sculpting treatment inresponse to the feedback, and may optionally also modify the plannedsculpting based at least in part on the feedback. Measurement systemsare further described in U.S. Pat. No. 6,315,413, the full disclosure ofwhich is incorporated herein by reference.

Laser beam 14 may be adjusted to produce the desired sculpting using avariety of alternative mechanisms. The laser beam 14 may be selectivelylimited using one or more variable apertures. An exemplary variableaperture system having a variable iris and a variable width slit isdescribed in U.S. Pat. No. 5,713,892, the full disclosure of which isincorporated herein by reference. The laser beam may also be tailored byvarying the size and offset of the laser spot from an axis of the eye,as described in U.S. Pat. Nos. 5,683,379, 6,203,539, and 6,331,177, thefull disclosures of which are incorporated herein by reference.

Still further alternatives are possible, including scanning of the laserbeam over the surface of the eye and controlling the number of pulsesand/or dwell time at each location, as described, for example, by U.S.Pat. No. 4,665,913, the full disclosure of which is incorporated hereinby reference; using masks in the optical path of laser beam 14 whichablate to vary the profile of the beam incident on the cornea, asdescribed in U.S. Pat. No. 5,807,379, the full disclosure of which isincorporated herein by reference; hybrid profile-scanning systems inwhich a variable size beam (typically controlled by a variable widthslit and/or variable diameter iris diaphragm) is scanned across thecornea; or the like. The computer programs and control methodology forthese laser pattern tailoring techniques are well described in thepatent literature.

Additional components and subsystems may be included with laser system10, as should be understood by those of skill in the art. For example,spatial and/or temporal integrators may be included to control thedistribution of energy within the laser beam, as described in U.S. Pat.No. 5,646,791, the full disclosure of which is incorporated herein byreference. Ablation effluent evacuators/filters, aspirators, and otherancillary components of the laser surgery system are known in the art.Further details of suitable systems for performing a laser ablationprocedure can be found in commonly assigned U.S. Pat. Nos. 4,665,913,4,669,466, 4,732,148, 4,770,172, 4,773,414, 5,207,668, 5,108,388,5,219,343, 5,646,791 and 5,163,934, the complete disclosures of whichare incorporated herein by reference. Basis data can be furthercharacterized for particular lasers or operating conditions, by takinginto account localized environmental variables such as temperature,humidity, airflow, and aspiration.

FIG. 2 is a simplified block diagram of an exemplary computer system 22that may be used by the laser surgical system 10 of the presentinvention. Computer system 22 typically includes at least one processor52 which may communicate with a number of peripheral devices via a bussubsystem 54. These peripheral devices may include a storage subsystem56, comprising a memory subsystem 58 and a file storage subsystem 60,user interface input devices 62, user interface output devices 64, and anetwork interface subsystem 66. Network interface subsystem 66 providesan interface to outside networks 68 and/or other devices, such as thewavefront measurement system 30.

User interface input devices 62 may include a keyboard, pointing devicessuch as a mouse, trackball, touch pad, or graphics tablet, a scanner,foot pedals, a joystick, a touchscreen incorporated into the display,audio input devices such as voice recognition systems, microphones, andother types of input devices. User input devices 62 will often be usedto download a computer executable code from a tangible storage media 29embodying any of the methods of the present invention. In general, useof the term “input device” is intended to include a variety ofconventional and proprietary devices and ways to input information intocomputer system 22.

User interface output devices 64 may include a display subsystem, aprinter, a fax machine, or non-visual displays such as audio outputdevices. The display subsystem may be a cathode ray tube (CRT), aflat-panel device such as a liquid crystal display (LCD), a projectiondevice, or the like. The display subsystem may also provide a non-visualdisplay such as via audio output devices. In general, use of the term“output device” is intended to include a variety of conventional andproprietary devices and ways to output information from computer system22 to a user.

Storage subsystem 56 can store the basic programming and data constructsthat provide the functionality of the various embodiments of the presentinvention. For example, a database and modules implementing thefunctionality of the methods of the present invention, as describedherein, may be stored in storage subsystem 56. These software modulesare generally executed by processor 52. In a distributed environment,the software modules may be stored on a plurality of computer systemsand executed by processors of the plurality of computer systems. Storagesubsystem 56 typically comprises memory subsystem 58 and file storagesubsystem 60.

Memory subsystem 58 typically includes a number of memories including amain random access memory (RAM) 70 for storage of instructions and dataduring program execution and a read only memory (ROM) 72 in which fixedinstructions are stored. File storage subsystem 60 provides persistent(non-volatile) storage for program and data files, and may includetangible storage media 29 (FIG. 1) which may optionally embody wavefrontsensor data, wavefront gradients, a wavefront elevation map, a treatmentmap, and/or an ablation table. File storage subsystem 60 may include ahard disk drive, a floppy disk drive along with associated removablemedia, a Compact Digital Read Only Memory (CD-ROM) drive, an opticaldrive, DVD, CD-R, CD-RW, solid-state removable memory, and/or otherremovable media cartridges or disks. One or more of the drives may belocated at remote locations on other connected computers at other sitescoupled to computer system 22. The modules implementing thefunctionality of the present invention may be stored by file storagesubsystem 60.

Bus subsystem 54 provides a mechanism for letting the various componentsand subsystems of computer system 22 communicate with each other asintended. The various subsystems and components of computer system 22need not be at the same physical location but may be distributed atvarious locations within a distributed network. Although bus subsystem54 is shown schematically as a single bus, alternate embodiments of thebus subsystem may utilize multiple busses.

Computer system 22 itself can be of varying types including a personalcomputer, a portable computer, a workstation, a computer terminal, anetwork computer, a control system in a wavefront measurement system orlaser surgical system, a mainframe, or any other data processing system.Due to the ever-changing nature of computers and networks, thedescription of computer system 22 depicted in FIG. 2 is intended only asa specific example for purposes of illustrating one embodiment of thepresent invention. Many other configurations of computer system 22 arepossible having more or less components than the computer systemdepicted in FIG. 2.

Referring now to FIG. 3, one embodiment of a wavefront measurementsystem 30 is schematically illustrated in simplified form. In verygeneral terms, wavefront measurement system 30 is configured to senselocal slopes of a gradient map exiting the patient's eye. Devices basedon the Hartmann-Shack principle generally include a lenslet array tosample the gradient map uniformly over an aperture, which is typicallythe exit pupil of the eye. Thereafter, the local slopes of the gradientmap are analyzed so as to reconstruct the wavefront surface or map.

More specifically, one wavefront measurement system 30 includes an imagesource 32, such as a laser, which projects a source image throughoptical tissues 34 of eye E so as to form an image 44 upon a surface ofretina R. The image from retina R is transmitted by the optical systemof the eye (e.g., optical tissues 34 ) and imaged onto a wavefrontsensor 36 by system optics 37. The wavefront sensor 36 communicatessignals to a computer system 22′ for measurement of the optical errorsin the optical tissues 34 and/or determination of an optical tissueablation treatment program. Computer 22′ may include the same or similarhardware as the computer system 22 illustrated in FIGS. 1 and 2.Computer system 22′ may be in communication with computer system 22 thatdirects the laser surgery system 10, or some or all of the components ofcomputer system 22, 22′ of the wavefront measurement system 30 and lasersurgery system 10 may be combined or separate. If desired, data fromwavefront sensor 36 may be transmitted to a laser computer system 22 viatangible media 29, via an I/O port, via an networking connection 66 suchas an intranet or the Internet, or the like.

Wavefront sensor 36 generally comprises a lenslet array 38 and an imagesensor 40. As the image from retina R is transmitted through opticaltissues 34 and imaged onto a surface of image sensor 40 and an image ofthe eye pupil P is similarly imaged onto a surface of lenslet array 38,the lenslet array separates the transmitted image into an array ofbeamlets 42, and (in combination with other optical components of thesystem) images the separated beamlets on the surface of sensor 40.Sensor 40 typically comprises a charged couple device or “CCD,” andsenses the characteristics of these individual beamlets, which can beused to determine the characteristics of an associated region of opticaltissues 34. In particular, where image 44 comprises a point or smallspot of light, a location of the transmitted spot as imaged by a beamletcan directly indicate a local gradient of the associated region ofoptical tissue.

Eye E generally defines an anterior orientation ANT and a posteriororientation POS. Image source 32 generally projects an image in aposterior orientation through optical tissues 34 onto retina R asindicated in FIG. 3. Optical tissues 34 again transmit image 44 from theretina anteriorly toward wavefront sensor 36. Image 44 actually formedon retina R may be distorted by any imperfections in the eye's opticalsystem when the image source is originally transmitted by opticaltissues 34. Optionally, image source projection optics 46 may beconfigured or adapted to decrease any distortion of image 44.

In some embodiments, image source optics 46 may decrease lower orderoptical errors by compensating for spherical and/or cylindrical errorsof optical tissues 34. Higher order optical errors of the opticaltissues may also be compensated through the use of an adaptive opticelement, such as a deformable mirror (described below). Use of an imagesource 32 selected to define a point or small spot at image 44 uponretina R may facilitate the analysis of the data provided by wavefrontsensor 36. Distortion of image 44 may be limited by transmitting asource image through a central region 48 of optical tissues 34 which issmaller than a pupil 50, as the central portion of the pupil may be lessprone to optical errors than the peripheral portion. Regardless of theparticular image source structure, it will be generally be beneficial tohave a well-defined and accurately formed image 44 on retina R.

In one embodiment, the wavefront data may be stored in a computerreadable medium 29 or a memory of the wavefront sensor system 30 in twoseparate arrays containing the x and y wavefront gradient valuesobtained from image spot analysis of the Hartmann-Shack sensor images,plus the x and y pupil center offsets from the nominal center of theHartmann-Shack lenslet array, as measured by the pupil camera 51 (FIG.3) image. Such information contains all the available information on thewavefront error of the eye and is sufficient to reconstruct thewavefront or any portion of it. In such embodiments, there is no need toreprocess the Hartmann-Shack image more than once, and the data spacerequired to store the gradient array is not large. For example, toaccommodate an image of a pupil with an 8 mm diameter, an array of a20×20 size (i.e., 400 elements) is often sufficient. As can beappreciated, in other embodiments, the wavefront data may be stored in amemory of the wavefront sensor system in a single array or multiplearrays.

While the methods of the present invention will generally be describedwith reference to sensing of an image 44, it should be understood that aseries of wavefront sensor data readings may be taken. For example, atime series of wavefront data readings may help to provide a moreaccurate overall determination of the ocular tissue aberrations. As theocular tissues can vary in shape over a brief period of time, aplurality of temporally separated wavefront sensor measurements canavoid relying on a single snapshot of the optical characteristics as thebasis for a refractive correcting procedure. Still further alternativesare also available, including taking wavefront sensor data of the eyewith the eye in differing configurations, positions, and/ororientations. For example, a patient will often help maintain alignmentof the eye with wavefront measurement system 30 by focusing on afixation target, as described in U.S. Pat. No. 6,004,313, the fulldisclosure of which is incorporated herein by reference. By varying aposition of the fixation target as described in that reference, opticalcharacteristics of the eye may be determined while the eye accommodatesor adapts to image a field of view at a varying distance and/or angles.

The location of the optical axis of the eye may be verified by referenceto the data provided from a pupil camera 52. In the exemplaryembodiment, a pupil camera 52 images pupil 50 so as to determine aposition of the pupil for registration of the wavefront sensor datarelative to the optical tissues.

An alternative embodiment of a wavefront measurement system isillustrated in FIG. 3A. The major components of the system of FIG. 3Aare similar to those of FIG. 3. Additionally, FIG. 3A includes anadaptive optical element 53 in the form of a deformable mirror. Thesource image is reflected from deformable mirror 98 during transmissionto retina R, and the deformable mirror is also along the optical pathused to form the transmitted image between retina R and imaging sensor40. Deformable mirror 98 can be controllably deformed by computer system22 to limit distortion of the image formed on the retina or ofsubsequent images formed of the images formed on the retina, and mayenhance the accuracy of the resultant wavefront data. The structure anduse of the system of FIG. 3A are more fully described in U.S. Pat. No.6,095,651, the full disclosure of which is incorporated herein byreference.

The components of an embodiment of a wavefront measurement system formeasuring the eye and ablations comprise elements of a VISX WaveScan®,available from VISX, INCORPORATED of Santa Clara, Calif. One embodimentincludes a WaveScan® with a deformable mirror as described above. Analternate embodiment of a wavefront measuring system is described inU.S. Pat. No. 6,271,915, the full disclosure of which is incorporatedherein by reference.

FIG. 4 schematically illustrates a simplified set of modules, or acorrection system 100, for carrying out a method according to oneembodiment of the present invention. Correction system 100 can beintegrated or interfaced with, for example, computer system 22, orotherwise used in conjunction with laser surgical system 10. The modulesmay be software modules on a computer readable medium that is processedby processor 52 (FIG. 2), hardware modules, or a combination thereof.Any of a variety of commonly used platforms, such as Windows, MacIntosh,and Unix, along with any of a variety of commonly used programminglanguages, may be used to implement the present invention.

Correction system 100 can be configured to generate a refractivetreatment shape 110 for ameliorating a vision condition in a patient. Aninput module 102 typically receives a measurement surface aberration120, such as wavefront aberration data from wavefront sensors, whichcharacterize aberrations and other optical characteristics of the entireoptical tissue system imaged. Often, the wavefront aberrationcorresponds to a measurement surface that is disposed at or near a pupilplane of the eye. The data from the wavefront sensors are typicallygenerated by transmitting an image (such as a small spot or point oflight) through the optical tissues, as described above. Measurementsurface aberration 120 can include an array of optical gradients or agradient map.

Correction system 100 can include a transformation module 104 thatderives a treatment surface aberration. The treatment surface aberrationcan correspond to a treatment surface that is disposed at or near ananterior corneal surface of the eye, or a treatment surface thatcorresponds to a spectacle plane of the eye. Relatedly, the treatmentsurface may be disposed posterior to a pupil plane of the eye. Often,the treatment surface aberration is derived from measurement surfaceaberration 120 using a difference between the measurement surface andthe treatment surface. For example, optical gradient data from inputmodule 102 may be transmitted to transformation module 104, where atreatment surface aberration is mathematically reconstructed based onthe optical gradient data.

Correction system 100 can include an output module 106, such that thetreatment surface aberration generated by transformation module 104 canthen be transmitted to output module 106 where a refractive treatmentshape 110 can be generated based on the treatment surface aberration.Refractive treatment shape 110 may be transmitted to a laser treatmentapparatus for generation of a laser ablation treatment for the patient.Similarly, refractive treatment shape 110 may form the basis forfabrication of contact lenses, spectacles, or intra-ocular lenses.

As can be appreciated, the present invention should not be limited tothe order of steps, or the specific steps illustrated, and variousmodifications to the method, such as having more or less steps, may bemade without departing from the scope of the present invention.

In one embodiment, the present invention provides a method ofdetermining a refractive treatment shape for ameliorating a visioncondition in a patient. FIG. 5 depicts the steps of an exemplary methodaccording to the present invention. The refractive treatment shape canbe based on a treatment surface aberration that is derived from ameasurement surface aberration.

I. Measurement Surface Aberration

In general terms, a measurement surface aberration can be determinedfrom optical data corresponding to a measurement surface. For example, ameasurement surface aberration can be determined by measuring awavefront aberration of an eye of a patient. A wavefront measurementsystem that includes a wavefront sensor (such as a Hartmann-Shacksensor) may be used to obtain one or more measurement surfaceaberrations (e.g. wavefront maps) of the optical tissues of the eye. Thewavefront map may be obtained by transmitting an image through theoptical tissues of the eye and sensing the exiting wavefront surface.From the wavefront map, it is possible to calculate a surface gradientor gradient map across the optical tissues of the eye. A gradient mapmay comprise an array of the localized gradients as calculated from eachlocation for each lenslet of the Hartmann-Shack sensor.

A. Measurement Surface

There are a variety of devices and methods for measuring surfacecharacteristics of optical systems. The category of aberroscopes oraberrometers includes classical phoropter and wavefront approaches.Classical phoropters can be used to record optical data corresponding toa measurement surface that is disposed anterior to the cornea of an eye.For example, phoropters can measure low order aberrations at a distanceof about 12.5 mm anterior to the corneal surface. In many cases, thiswill correspond to a spectacle plane of the eye. Wavefront devices areoften used to measure both low order and high order aberrations adjacenta pupil plane, which can be about 3.5 mm posterior to the cornealsurface. Another category of measuring approaches includes topographybased measuring devices and methods. Topography typically involvesproviding optical data corresponding to a measurement surface that isdisposed at or near the corneal surface of the eye.

B. Aberration

As noted above, the measurement surface aberration can be based on arefractive measurement as determined by an optometer, or any of a widevariety of devices for obtaining irregular aberration data. Similarly,the measurement surface aberration can be a measurement surfacewavefront map, as determined by a wavefront measurement device. What ismore, the measurement surface aberration may reflect both low order andhigh order aberrations of the eye of a patient.

II. Treatment Surface Aberration

When a measurement surface aberration of an optical system has beendetermined, it is then possible to derive a treatment surface aberrationof the optical system. In the case of refractive surgical methods, forexample, a treatment surface aberration corresponding to a corneal planecan be derived from a measurement surface aberration as determined in aplane other than the corneal plane.

A. Treatment Surface

The treatment surface aberration corresponds to a treatment surface,which is typically disposed at or near an anterior surface of a corneaof an eye. Often, the treatment surface will correspond to a cornealplane associated with the eye, as in the case of ablative laser eyesurgery or contact lens treatments. At other times, the treatmentsurface may correspond to a spectacle plane associated with the eye, asin the case of spectacle treatments. Further, the treatment surface canbe posterior to the pupil plane of the eye, as in the case ofintraocular lens treatments.

B. Derivation of Treatment Surface Aberration

The treatment surface aberration can be derived from the measurementsurface aberration, based on a difference between the measurementsurface and the treatment surface. The difference between themeasurement surface and the treatment surface, for example, can includea distance measurement that represents a difference between the twosurfaces. In some embodiments, the distance measurement is based on avertex distance difference, the vertex distance difference reflecting adistance between a vertex of the measurement surface and a vertex of thetreatment surface.

1. Classical Vertex Correction Formulas

Traditionally, the power of a lens is measured in diopters, and can bedefined as the reciprocal of the lens focal length in meters. FIG. 6shows a schematic diagram of an optical system. The system includes afirst plane disposed at a first distance from a focal plane, the firstdistance corresponding to a first focal length, and a second planedisposed at a second distance from the focal plane, the second distancecorresponding to a second focal length. Although the first and secondplanes are illustrated as flat surfaces, these planes can also representcurved surfaces such as lenses, wavefronts, and other representations ofoptical surfaces. In the exemplary optical system depicted by FIG. 6legend (a), the focal plane may be associated with a retinal plane, thefirst plane may be associated with a spectacle plane, and the secondplane may be associated with a corneal plane.

A treatment surface can correspond to, or be based upon, a spectaclesurface, corneal surface, pupil surface, and the like. A spectaclesurface is typically about 12.5 mm anterior to the cornea of the eye. Apupil surface or plane is typically about 3.5 mm posterior to the corneaof the eye. An intraocular lens surface is usually about 0.5 mmposterior to the pupil surface or plane of the eye. A contact lenssurface is typically at or slightly anterior to the cornea of the eye.If the treatment surface and the measurement surface are substantiallyin the same plane, there may be no need for a vertex correction.

When prescribing spectacles, for example, an optometrist may first makeor consider an aberration measurement such as a refractive measurementof the eye, where the aberration measurement corresponds to ameasurement surface at or near a pupil plane or surface of the eye.Because the treatment surface may not be the same as the measurementsurface, it is often desirable to make a power adjustment in order todetermine the corrective surface shape or treatment surface aberration.In the case of spectacles, the treatment surface is disposed anterior tothe corneal surface, usually by about 12.5 mm.

Likewise, when prescribing contact lenses, an optometrist can consider arefractive correction corresponding to the spectacle plane, and make apower adjustment to account for the difference between the spectacleplane and the corneal plane. In this case, the adjustment often dependson a vertex distance, corresponding to the distance between theposterior surface of the spectacle lens and the anterior surface of thecornea.

Thus, given a measurement surface aberration, it is possible to derive atreatment surface aberration based on a difference between the treatmentsurface and the measurement surface. Often, the difference will be avertex distance between the treatment surface and the measurementsurface. As further discussed below, the treatment surface aberrationcan then be used to determine a refractive treatment shape. In the caseof corrective spectacles, the refractive treatment shape can be a basisfor a prescription for the patient, where the treatment shapecorresponds to the spectacle plane or surface.

Typically, the measurement surface aberration corresponds to a firstpower data, and the treatment surface aberration corresponds to aderived second power data. The second power data can be derived from thefirst power data and the distance between the measurement surface andthe treatment. To achieve the effect of a power change, in terms of avertex correction, a vertex distance measure can be based on adifference between the measurement surface and the treatment surface.The vertex correction represents a power change between the first powerdata and the second power data. In this sense, the derivation of thesecond power corresponds to a vertex correction of the first power. Thevertex of a lens curve can be defined as the apex of the lens curve, oras the point on the lens curve in which the lens curve axis intersectsit.

a. Traditional (Non Wavefront)

Traditional phoropters can be used to make traditional opticalaberration measurements such as sphere and cylinder. Such aberrationmeasurements are often expressed in terms of dioptric power. Referringagain to FIG. 6 legend (a), assuming the power corresponding to thesecond plane, e.g. a corneal plane, is S, and the power correspondingthe first plane, e.g. a spectacle plane, is S′, it is possible todescribe the relationship between the two powers with the followingequations.

$\begin{matrix}{{S = \frac{1}{f}},} & (1) \\\begin{matrix}{S^{\prime} = \frac{1}{f + d}} \\{= \frac{S}{1 + {dS}}}\end{matrix} & (2)\end{matrix}$

Power can be expressed in units of diopters. f represents the distancebetween the focal plane and the second plane, although here this term isnot a factor in the relationship between the two power measurements Sand S′. d represents the vertex distance between the first and secondplanes. Where a first plane treatment surface is disposed anterior to asecond plane measurement surface, d will typically have a positivevalue. For example, for spectacle treatments, d can be about 0.0125 m,and for refractive surgery treatments, d can be about 0.0035 m.Conversely, where the first plane treatment shape is disposed posteriorto a second plane measurement surface, d will typically have a negativevalue. For example, for intraocular lens treatments, d can be about−0.0005 m.

Sphere is a low order aberration corresponding to defocus, and cylinderis a low order aberration corresponding to astigmatism. To consider acombination of sphere and cylinder powers, it is possible to replace Sby (S+C) where C stands for cylinder power at the maximum meridian.Thus, cylinder at the spectacle plane can be represented by C′, where

$\begin{matrix}{C^{\prime} = {\frac{S + C}{1 + {d\left( {S + C} \right)}} - {S^{\prime}.}}} & (3)\end{matrix}$

These formulae can be used to calculate the power change associated witha vertex distance.

b. Wavefront

In addition to the traditional phoropter approaches discussed above, itis also possible to evaluate optical systems based on wavefrontanalysis. Wavefront analysis can be useful in evaluating low order andhigh order aberrations. Referring again to FIG. 6, it is possible toconsider the first and second planes as associated with a generalwavefront. The wavefront can begin at a virtual focal pointcorresponding to the focal plane, and propagate from plane two towardplane one. For each point along the wavefront surface, a local slope canbe calculated. For example, the local slope can be the first derivativeof the surface at a certain point. The local slope reflects a surfacevalue at that point, as well as the surface values of the surroundingpoints. The local slope can be a direction-dependent vector. Because thewavefront local slopes are proportional to the local focal length, asthe wavefront is propagated forward, the slope of the wavefront can bescaled by a factor of a such that:

$\begin{matrix}{\alpha = \frac{f}{f + d}} & (4)\end{matrix}$where f is the focal length of the wavefront and d is the vertexdistance. Here, the vertex distance can represent a difference betweenthe measurement surface, or plane two, and the treatment surface, orplane one. Thus, by making an initial measurement of the wavefront atplane two, it is possible to calculate a new wavefront surface at planewhere individual points on the new surface have a local curvature thatis derived by the scaling factor as discussed above. In the exemplaryoptical system depicted by FIG. 6 legend (b), the first plane canrepresent a corneal plane, the second plane can represent a pupil plane,and the focal plane can represent a retinal plane. If the treatmentsurface is anterior to the measurement surface, then the vertex distanceis positive, and if the treatment surface is posterior to themeasurement surface, then the vertex distance is negative. Similarly,for the myopia case, because the power is negative, the focal lengthcould take a negative value. Generally a can have a positive value, asthe absolute value of f is often much larger than d.

As discussed above, vertex correction can be used with traditionalaberrometry approaches. It is also possible to use vertex correctionwith wavefront approaches. Here, W(x,y) represents the wavefront at themeasurement plane and W′(x,y) represents the wavefront at the treatmentplane with vertex distance of d. The local slope is assumed to bescaled, as discussed above. Thus, the following equations are partialderivatives of the corrected wavefront at the treatment plane.

$\begin{matrix}{{\frac{\partial W^{\prime}}{\partial x} = {\frac{f}{f + d}\frac{\partial W}{\partial x}}}{\frac{\partial W^{\prime}}{\partial y} = {\frac{f}{f + d}\frac{\partial W}{\partial y}}}} & (5)\end{matrix}$

It can be demonstrated that the classical formula for vertex correctionholds with the assumption that the local slopes can be scaled accordingto a scaling factor of f/(f+d). The following examples illustrate thisprinciple with respect to (i) sphere, or defocus, (ii) cylinder, orastigmatism, (iii) coma, and (iv) spherical aberration. Wavefronts canbe expressed in terms of polynomial equations. This equation is usefulfor the derivations that follow.

$\begin{matrix}{\frac{\partial^{2}W^{\prime}}{\partial r^{2}} = {{\frac{x^{2}}{x^{2} + y^{2}}\frac{\partial^{2}W^{\prime}}{\partial x^{2}}} + {\frac{2{xy}}{x^{2} + y^{2}}\frac{\partial^{2}W^{\prime}}{{\partial x}{\partial y}}} + {\frac{y^{2}}{x^{2} + y^{2}}{\frac{\partial^{2}W^{\prime}}{\partial y^{2}}.}}}} & (6)\end{matrix}$

(i) Sphere

In the following discussion, Zernike polynomials are used to representthe ocular aberrations. Starting with a sphere, where W(r)=c₂ ⁰√{squareroot over (3)}(2r²−1), corresponding to the wavefront at the secondplane, the curvature of the converted wavefront W′(r) at the first planecan be expressed as

$\begin{matrix}\begin{matrix}{\frac{\partial^{2}W^{\prime}}{\partial r^{2}} = {{\frac{x^{2}}{x^{2} + y^{2}}\frac{\partial^{2}W^{\prime}}{\partial x^{2}}} + {\frac{2{xy}}{x^{2} + y^{2}}\frac{\partial^{2}W^{\prime}}{{\partial x}{\partial y}}} + {\frac{y^{2}}{x^{2} + y^{2}}\frac{\partial^{2}W^{\prime}}{\partial y^{2}}}}} \\{{= {4\sqrt{3}c_{2}^{0}\frac{f}{f + d}}},{or}}\end{matrix} & (7) \\{{\frac{\partial^{2}W^{\prime}}{\partial r^{2}} = {4\sqrt{3}c_{2}^{0}\frac{f}{f + d}}},} & (8)\end{matrix}$where the curvature of the vertex corrected wavefront can be calculatedusing Equation (6). Here, r represents the normalized pupil radius withvalues from 0 to 1, x and y are the normalized values in x- and y-axis,f is the local focal length, or the reciprocal of local power, and c₂ ⁰is the Zernike coefficient of defocus term. From the definition ofpower, we have

$\begin{matrix}{{\frac{\partial^{2}W}{\partial r^{2}} = {4\sqrt{3}c_{2}^{0}}}{S = {{\frac{1}{R^{2}}{\frac{\partial^{2}W}{\partial r^{2}}.S^{\prime}}} = {\frac{1}{R^{2}}\frac{\partial^{2}W^{\prime}}{\partial r^{2}}}}}} & (9)\end{matrix}$From Equations (8) and (9), we obtain the following formula

$\begin{matrix}{S^{\prime} = {{\frac{f}{f + d}\mspace{14mu} S} = {\frac{S}{1 + {Sd}}.}}} & (10)\end{matrix}$

Equation (10) is the classical formula for vertex correction of puresphere power, thus demonstrating that vertex correction can beeffectively used in wavefront analysis.

(ii) Cylinder

In another example for astigmatism, W(r, θ)=c₂ ² √{square root over(6)}r² sin 2θ+c₂ ²√{square root over (6)}r² cos 2θ corresponds to thewavefront at the second plane, a similar approach can be used to obtainthe curvature of the corrected wavefront as

$\begin{matrix}\begin{matrix}{\frac{\partial^{2}W^{\prime}}{\partial r^{2}} = {\left( {{2\sqrt{6}c_{2}^{- 2}\sin\; 2\theta} + {2\sqrt{6}c_{2}^{2}\cos\; 2\theta}} \right)\frac{f}{f + d}}} \\{= {\frac{\partial^{2}W}{\partial r^{2}}{\frac{f}{f + d}.}}}\end{matrix} & (11)\end{matrix}$Denoting P′ and P as the curvatures of W′ (converted wavefront) and W(measured wavefront) respectively,

$\begin{matrix}\begin{matrix}{P^{\prime} = {P\;\frac{f}{f + d}}} \\{= {\frac{P}{1 + {Pd}}.}}\end{matrix} & (12)\end{matrix}$By replacing P with S+C, it is possible to obtain the classical vertexcorrection for cylinder

$\begin{matrix}{C^{\prime} = {\frac{S + C}{1 + {d\left( {S + C} \right)}} - {S^{\prime}.}}} & (13)\end{matrix}$

(iii) Coma

In addition to the low order wavefront vertex corrections discussedabove, it is also possible to use vertex correction with wavefrontmeasurements that include high order aberrations. For example,horizontal coma can be expressed as W(r, θ)=√{square root over (8)}c₃¹(3r³−2r)cos θ, again corresponding to the wavefront at the secondplane. With an approach similar to that described above, it is possibleto calculate the derivatives to x and to y and then calculate thecurvature to r as

$\begin{matrix}\begin{matrix}{\frac{\partial^{2}W^{\prime}}{\partial r^{2}} = {\frac{f}{f + d}18\sqrt{8}c_{3}^{1}x}} \\{= {\frac{\partial^{2}W}{\partial r^{2}}{\frac{f}{f + d}.}}}\end{matrix} & (14)\end{matrix}$Denoting P′ and P as the curvatures of W′ (converted wavefront) and W(measured wavefront) respectively,

$\begin{matrix}\begin{matrix}{P^{\prime} = {P\;\frac{f}{f + d}}} \\{= {\frac{P}{1 + {Pd}}.}}\end{matrix} & (15)\end{matrix}$

(iv) Spherical Aberrations

In another example, a spherical aberration can be expressed asW(r)=√{square root over (5)}c₄ ⁰(6r⁴−6r²+1). Again, with an approachsimilar to that described above, it is possible to calculate thederivatives to x and to y and then calculate the curvature to r todetermine the curvature of the corrected wavefront as

$\begin{matrix}\begin{matrix}{\frac{\partial^{2}W^{\prime}}{\partial r^{2}} = {\frac{f}{f + d}\left( {{72r^{2}} - 12} \right)\sqrt{5}c_{4}^{0}}} \\{= {\frac{\partial^{2}W}{\partial r^{2}}{\frac{f}{f + d}.}}}\end{matrix} & (16)\end{matrix}$Denoting P′ and P as the curvatures of W′ (converted wavefront) and W(measured wavefront) respectively,

$\begin{matrix}\begin{matrix}{P^{\prime} = {P\;\frac{f}{f + d}}} \\{= {\frac{P}{1 + {Pd}}.}}\end{matrix} & (17)\end{matrix}$

Therefore, for low order aberrations as well as for high orderaberrations, it can be shown that by means of a slope scaling as appliedin wavefront, it is possible to achieve the effect of power change asdefined in a classical sense. Such approaches can be useful indetermining treatment surface aberrations based on measurement surfaceaberrations.

2. New Algorithm for Vertex Correction

Treatment surface aberrations can also be determined based on variousalgorithmic approaches. In some embodiments, the treatment surfaceaberration is a treatment surface wavefront map. The treatment surfacewavefront map can be derived at least in part by local slope scaling ofa measurement surface wavefront map. For example, a treatment surfacewavefront map can be derived at least in part by applying a scalingfactor of 1/(1+Pd) to a slope of a measurement surface wavefront map,where P represents a local curvature of the measurement surfacewavefront map and d represents a difference between a measurementsurface and a treatment surface. For example, P can be based on a secondderivative of the measurement surface wavefront map. P can also be basedon a pupil radius of the eye. The following examples illustratealgorithmic approaches that incorporate such principles.

a. Constant HOA

This algorithm assumes that the average curvature for low orderaberrations (LOA), as manifested by sphere and cylinder power terms, isaffected by vertex distance change. High order aberrations (HOA) areconsidered as local irregularity add-ons to the mean curvature, and arenot affected by vertex distance change. Thus, a total wavefront map canbe separated into low order and high order portions as shown by thefollowing formulaW(x,y)=W _(L)(x,y)+W _(H)(x,y).  (18)

For the low order portion, it is possible to obtain the sphere andcylinder power terms by means of a Zernike decomposition method[S,C]=ZD[W _(L)(x,y)],  (19)where S and C represent the sphere and cylinder power terms,respectively, and ZD represents a Zernike decomposition operator. Thevertex corrected sphere S′ and cylinder C′ power terms can be obtainedfrom the following formulae

$\begin{matrix}{{S^{\prime} = \frac{S}{1 + {dS}}},} & (20) \\{C^{\prime} = {\frac{S + C}{1 + {d\left( {S + C} \right)}} - {S^{\prime}.}}} & (21)\end{matrix}$

The vertex corrected wavefront can then be obtained by adding theuncorrected high order portion of the original wavefront with theZernike expansion operator applied to the corrected sphere S′ andcylinder C′ asW′(x,y)=ZE(S′,C′)+W _(H)(x,y),  (22)where ZE stands for a Zernike expansion operator.

b. Varying HOA

This algorithm segments the wavefront measurement into multipleportions, and is designed to have each portion of the correctedwavefront focused on or toward the focal point of the optical system,regardless of the wavefront shape. Thus, the local slope of each portionof the wavefront measurement can be scaled by a factor of f/(f+d) wheref represents the local focal length and d represents the vertexdistance. By applying the following algorithms, it is possible to obtainthe vertex corrected wavefront:

-   -   1. Calculate x- and y-gradient by the following algorithm:        -   Along the x axis:        -   a. ∂W/∂x=[W(i, j+1)−W(i, j)]/dx if [i,j] lands on left edge        -   b. ∂W/∂x=[W(i, j)−W(i, j−1)]/dx if [i,j] lands on right edge        -   c. ∂W/∂x=[W(i, j +1)−W(i, j−1)]/2dx otherwise within pupil        -   Along the y axis:        -   d. ∂W/∂y=[W(i, j)−W(i+1, j)]/dy if [i,j] lands on upper edge        -   e. ∂W/∂y=[W(i−1, j)−W(i, j)]/dy if [i,j] lands on lower edge        -   f. ∂W/∂y=[W(i−1, j)−W(i+1, j)]/2dy otherwise within pupil        -   If [i,j] is outside the pupil, the data is not considered.    -   2. Calculate local curvature P using this algorithm:

${{a.\mspace{14mu}{Calculate}}\mspace{14mu}\frac{\partial^{2}W}{\partial x^{2}}},{\frac{\partial^{2}W}{\partial y^{2}}\mspace{14mu}{and}\mspace{14mu}\frac{\partial^{2}W}{{\partial x}{\partial y}}\mspace{14mu}{from}\mspace{14mu}\frac{\partial W}{\partial x}\mspace{14mu}{and}\mspace{14mu}\frac{\partial W}{\partial y}}$using  algorithm  1.${b.\mspace{14mu}\frac{\partial^{2}W}{\partial r^{2}}} = {{\frac{x^{2}}{x^{2} + y^{2}}\frac{\partial^{2}W}{\partial x^{2}}} + {\frac{2{xy}}{x^{2} + y^{2}}\frac{\partial^{2}W}{{\partial x}{\partial y}}} + {\frac{y^{2}}{x^{2} + y^{2}}\frac{\partial^{2}W}{\partial y^{2}}}}$${{c.\mspace{14mu}{Calculate}}\mspace{14mu}{local}\mspace{14mu}{curvature}\mspace{14mu} P} = {\frac{1}{R^{2}}\frac{\partial^{2}W}{\partial r^{2}}\left( {R\mspace{14mu}{being}\mspace{14mu}{pupil}\mspace{14mu}{radius}} \right)}$

-   -   3. Scale the wavefront local curvature with this algorithm:

$\frac{\partial W^{\prime}}{\partial x} = {\frac{1}{1 + {Pd}}\frac{\partial W}{\partial x}}$$\frac{\partial W^{\prime}}{\partial y} = {\frac{1}{1 + {Pd}}\frac{\partial W}{\partial y}}$

-   -   4. Reconstruct the corrected wavefront W′(x,y) with this        algorithm:        -   a. Calculate Fourier transform of ∂W′/∂x and ∂W′/∂y to get            c_(u) and c_(v), respectively.        -   b. Multiply u with c_(u) and v with c_(v) and divide by            u²+v².        -   c. Inverse Fourier transform to get W′(x,y).        -   d. Calculate ∂W′/∂x and ∂W′/∂y using algorithm 1, adjusted            with the edge being the entire frame as oppose to pupil            edge.        -   e. Replace ∂W′/∂x and ∂W′/∂y with values from step 3 within            the pupil, leave values outside pupil untouched.        -   f. Determine if a predefined criteria is met, or if a            predetermined number of iterations have been completed. If            not, go to step (a) and repeat through step (f).        -   g. Provide an estimate of W′(x,y).

A predefined criteria of step (f) could be, for example, the RMS errorof the reconstructed wavefront based on a comparison between W′_(i) andW′_(i−1), in the ith and (i−1)th iterations, respectively.Alternatively, other optical quality gauges may be used. In oneembodiment, the predetermined number of iterations in step (f) is 10. Asillustrated in the above algorithm, it is possible to derive a treatmentsurface wavefront map based on an iterative Fourier reconstructionalgorithm. Thus the entire algorithm, steps 1 to 4, uses both Fourierreconstruction (step 4) and local slope scaling (step 3).

The theory behind Fourier reconstruction can be described as follows.Suppose wavefront W(x,y) is expanded into Fourier series asW(x,y)=∫∫c(u,v) exp[i2π(ux+vy)]dudv,  (23)where c(u, v) is the expansion coefficient. Taking partial derivative tox and y, respectively in the above equation, provides

$\begin{matrix}\left\{ \begin{matrix}{\frac{\partial{W\left( {x,y} \right)}}{\partial x} = {{\mathbb{i}2\pi}{\int{\int{{{uc}\left( {u,v} \right)}{\exp\left\lbrack {{\mathbb{i}2\pi}\left( {{ux} + {vy}} \right)} \right\rbrack}{\mathbb{d}u}{\mathbb{d}v}}}}}} \\{\frac{\partial{W\left( {x,y} \right)}}{\partial y} = {{\mathbb{i}2\pi}{\int{\int{{{vc}\left( {u,v} \right)}{\exp\left\lbrack {{\mathbb{i}2\pi}\left( {{ux} + {vy}} \right)} \right\rbrack}{\mathbb{d}u}{\mathbb{d}v}}}}}}\end{matrix} \right. & (24)\end{matrix}$

Denoting c_(u) to be the Fourier transform of x-derivative of W(x,y) andc_(v) to be the Fourier transform of y-derivative of W(x,y), provides

$\begin{matrix}\left\{ \begin{matrix}{\frac{\partial{W\left( {x,y} \right)}}{\partial x} = {\int{\int{{c_{u}\left( {u,v} \right)}{\exp\left\lbrack {{\mathbb{i}2\pi}\left( {{ux} + {vy}} \right)} \right\rbrack}{\mathbb{d}u}{\mathbb{d}v}}}}} \\{\frac{\partial{W\left( {x,y} \right)}}{\partial y} = {\int{\int{{c_{v}\left( {u,v} \right)}{\exp\left\lbrack {{\mathbb{i}2\pi}\left( {{ux} + {vy}} \right)} \right\rbrack}{\mathbb{d}u}{\mathbb{d}v}}}}}\end{matrix} \right. & (25)\end{matrix}$Comparing these two sets of equations, provides

$\begin{matrix}\left\{ \begin{matrix}{{c_{u}\left( {u,v} \right)} = {{\mathbb{i}2}\;\pi\;{{uc}\left( {u,v} \right)}}} \\{{c_{v}\left( {u,v} \right)} = {{\mathbb{i}2}\;\pi\;{{vc}\left( {u,v} \right)}}}\end{matrix} \right. & (26)\end{matrix}$Combining these two equations with u multiplied in both sides of thefirst equation and v multiplied in both sides of the second equation,providesuc _(u)(u,v)+vc _(v)(u,v)=i2π(u ² +v ²)c(u,v).  (27)Therefore, the Fourier transform of wavefront can be obtained as

$\begin{matrix}{{c\left( {u,v} \right)} = {{- \frac{i\left\lbrack {{{uc}_{u}\left( {u,v} \right)} + {{vc}_{v}\left( {u,v} \right)}} \right\rbrack}{2{\pi\left( {u^{2} + v^{2}} \right)}}} = {- {\frac{i}{2{\pi\left( {u^{2} + v^{2}} \right)}}\left\lbrack {{u{\int{\int{\frac{\partial{W\left( {x,y} \right)}}{\partial x}\exp\left. \quad\left\lbrack {{- {\mathbb{i}2\pi}}\left( {{ux} + {vy}} \right.} \right. \right\rbrack}}}} + {v{\int{\int{\frac{\partial{W\left( {x,y} \right)}}{\partial y}{\exp\left\lbrack {- {{\mathbb{i}2\pi}\left( {{ux} + {vy}} \right)}} \right\rbrack}}}}}} \right\rbrack}}}} & (28)\end{matrix}$Hence, taking an inverse Fourier transform, it is possible to obtain thewavefront asW(x,y)=∫∫c(u,v) exp[i2π(ux+vy)]dudv.   (29)III. Refractive Treatment Shape

Once a treatment surface aberration has been derived by a method asdescribed above, it is possible to determine a prescription or arefractive treatment shape based on the treatment surface aberration.For example, a prescription can be derived for ameliorating a visioncondition in an eye of a patient. A refractive treatment shape can bedetermined based on the treatment surface aberration of the eye, and arefractive treatment shape can be embodied in any of a variety ofcorrective optical devices or procedures, including refractive lasersurgery, spectacles, contact lenses, intraocular lenses, and the like.

IV. Example: Evaluating Classical Formulas and New Algorithms

In some embodiments, it is useful to evaluate the convergence of Fourierreconstruction used in the vertex correction algorithms discussed above.Such approaches are discussed in commonly owned patent application Ser.No. 10/601,048 filed Jun. 20, 2003, the entirety of which is herebyincorporated by reference. It is also useful to evaluate the accuracy ofthe varying high order aberration algorithm as compared to the classicalformulas discussed above (i.e. sphere, sphere and cylinder). Forexample, one test is to show the comparison between the algorithmicapproaches and the traditional approaches for myopic, hyperopic, andastigmatism cases. FIG. 7 shows the comparison of vertex correctedsphere and cylinder using the varying high order aberration algorithmdescribed above as compared to classical formulas (i.e. sphere, sphereand cylinder) for (a) hyperopia+3D; (b) myopia−3D; (c)astigmatism−2DS/−1.5DC. It is quite clear that the results are verygood. Good results can be shown by a small error. For example, if thedifference is less than 0.05D, or smaller than 2.5%, it can generally beconsidered good. For pure sphere cases (e.g. myopia and hyperopia), theerror can be larger, due to coarse sampling of wavefront data in thecalculation.

For high order aberrations, it has been shown with two examples (i.e.coma, spherical aberrations) in theory that the vertex correctedwavefront follows the power relationship given by the classical formulaof vertex correction. FIG. 8 shows wavefront surface plots of apre-vertex correction (left panel) and post-vertex correction (rightpanel) corresponding to a 12.5 mm vertex correction as accomplished by avarying high order aberration algorithm.

In terms of the efficiency of a varying high order aberration algorithm,the following table shows the running time taken for such a vertexcorrection algorithm with respect to the number of iterations taken inthe Fourier reconstruction, corresponding to step 4 of the algorithm, ina 1.13 GHz laptop computer. With 10 iterations, the algorithm can takemore than 2 seconds in real time. Fortunately, this vertex correctionmay only be needed when a treatment table is generated, which in itselfmay take minutes. Treatment tables are files that can store commands fora laser to deliver individual laser pulses, in the context of a laserablation treatment. For example, the commands can be for laser pulseduration and size.

Iterations 1 2 5 10 20 50 200 Time (s) 0.340 0.521 1.231 2.303 4.25610.40 41.34

Thus in one embodiment, as part of the algorithm, Fourier reconstructioncan require about 10 iterations to achieve planned results given by100-micron sampling rate.

While the exemplary embodiments have been described in some detail, byway of example and for clarity of understanding, those of skill in theart will recognize that a variety of modification, adaptations, andchanges may be employed. Hence, the scope of the present inventionshould be limited solely by the appending claims.

1. A method of determining a refractive treatment shape for amelioratinga vision condition in a patient, the method comprising: a) measuring awavefront aberration of an eye of the patient, the wavefront aberrationcorresponding to a measurement surface that is disposed at or near apupil plane of the eye, in order to provide a measurement surfaceaberration comprising a measurement surface wavefront map; b) deriving atreatment surface aberration of the eye, the treatment surfaceaberration corresponding to a treatment surface of the eye andcomprising a treatment surface wavefront map derived at least in part bylocal slope scaling of the measurement surface wavefront map, thetreatment surface aberration derived from the measurement surfaceaberration using a difference between the measurement surface and thetreatment surface; and c) determining the refractive treatment shapebased on the treatment surface aberration of the eye.
 2. The method ofclaim 1, wherein the treatment surface is disposed at or near ananterior corneal surface of the eye.
 3. The method of claim 1, whereinthe treatment surface corresponds to a spectacle plane of the eye. 4.The method of claim 1, wherein the treatment surface is disposedposterior to a pupil plane of the eye.
 5. The method of claim 1, whereinthe treatment surface wavefront map is derived at least in part byapplying a scaling factor of 1/(1+Pd) to a slope of the measurementsurface wavefront map, where P represents a local curvature of themeasurement surface wavefront map and d represents a difference betweenthe measurement surface and the treatment surface.
 6. The method ofclaim 5, wherein P is based on a second derivative of the measurementsurface wavefront map.
 7. The method of claim 6, wherein P is based on apupil radius of the eye.
 8. The method of claim 7, wherein the treatmentsurface wavefront map is derived according to an iterative Fourierreconstruction algorithm.
 9. The method of claim 8, wherein theiterative Fourier reconstruction algorithm comprises 10 iterations. 10.The method of claim 1, wherein the measurement surface aberrationreflects low order and high order aberrations of the eye of the patient.11. The method of claim 1, wherein a difference between the measurementsurface and a retinal surface of the eye corresponds to a first vertexmeasure, and a difference between the treatment surface and the retinalsurface of the eye corresponds to a second vertex measure.
 12. A methodof ameliorating a vision condition in a patient, the method comprising:a) measuring a wavefront aberration of an eye of the patient, thewavefront aberration corresponding to a measurement surface that isdisposed at or near a pupil plane of the eye, in order to provide ameasurement surface aberration, wherein the measurement surfaceaberration comprises low order and high order aberrations of the eye ofthe patient; b) deriving a treatment surface aberration of the eye, thetreatment surface aberration corresponding to a treatment surface of theeye, the treatment surface aberration derived from the measurementsurface aberration using a difference between the measurement surfaceand the treatment surface; c) determining a refractive treatment shapebased on the treatment surface aberration of the eye; and d) applyingthe refractive treatment shape to the eye of the patient to amelioratethe vision condition.
 13. The method of claim 12, wherein the treatmentsurface is disposed at or near an anterior corneal surface of the eye,and the refractive treatment shape is applied to the eye of the patientin a treatment modality selected from the group consisting of: (i)ablating a corneal surface of the eye to provide a corneal surface shapethat corresponds to the refractive treatment shape, and (ii) providingthe patient with a contact lens that has a shape that corresponds to therefractive treatment shape.
 14. The method of claim 12, wherein thetreatment surface corresponds to a spectacle plane of the eye, and therefractive treatment shape is applied to the eye of the patient byproviding the patient with a spectacle lens that has a shape thatcorresponds to the refractive treatment shape.
 15. The method of claim12, wherein the treatment surface is disposed posterior to a pupil planeof the eye, and the refractive treatment shape is applied to the eye ofthe patient by providing the patient with an intra-ocular lens that hasa shape that corresponds to the refractive treatment shape.
 16. A systemfor generating a refractive treatment shape for ameliorating a visioncondition in an eye of a patient, the system comprising: a) an inputmodule that accepts a measurement surface aberration, the measurementsurface aberration based on a wavefront aberration the eye, thewavefront aberration corresponding to a measurement surface that isdisposed at or near a pupil plane of the eye, wherein a differencebetween the measurement surface and a retinal surface of the eyecorresponds to a first vertex measure; b) a transformation module thatderives a treatment surface aberration, the treatment surface aberrationcorresponding to a treatment surface of the eye, the treatment surfaceaberration derived from the measurement surface aberration using adifference between the measurement surface and the treatment surface,wherein a difference between the treatment surface and the retinalsurface of the eye corresponds to a second vertex measure; and c) anoutput module that generates the refractive treatment shape based on thetreatment surface aberration.
 17. The system of claim 16, wherein thetreatment surface is disposed at or near an anterior corneal surface ofthe eye.